Question

Interpreting Power For the sample data in Example 1 “Adult Sleep” from this section, Minitab and StatCrunch show that the hypothesis test has power of 0.4943 of supporting the claim that $\mu <7$ hours of sleep when the actual population mean is 6.0 hours of sleep. Interpret this value of the power, then identify the value of $\beta$and interpret that value. (For the t test in this section, a “noncentrality parameter” makes calculations of power much more complicated than the process described in Section 8-1, so software is recommended for power calculations.)

Short answer

If the true population mean sleep time is equal to 6.0 hours, there is a 49.43% chance of making the correct conclusion of rejecting the false null hypothesis/supporting the stated claim.

The value of $\beta$ is equal to 0.5057.

If the true population mean sleep time is equal to 6.0 hours, there is a 50.57% chance of making the incorrect conclusion of failing to reject the false null hypothesis/rejecting the stated claim.

Step-by-step solution

Given information

A sample of 12 adults is randomly selected. The actual population’s mean sleep time is equal to 6.0 hours. It is claimed that the mean sleep time is less than 7 hours.

Hypotheses

The appropriate hypotheses for testing the given claim are written as follows:

Null hypothesis: The mean sleep time is equal to 7.0 hours.

$H_0:\mu =7$

Alternative Hypothesis: The mean sleep time is less than 7.0 hours.

$H_0:\mu <7$

The test is left-tailed.

Power of the test

The power of the test refers to the probability of rejecting the null hypothesis when it is actually false. In simple words, the value of the power of the test explains the probability of making the correct conclusion of the claim.

Here, the value of the power of the test is equal to 0.4943.

The actual population’s mean sleep time is equal to 6.0 hours. It is claimed that the mean sleep time is less than 7 hours.

Thus, it means that there is a 49.43% probability of making the correct conclusion that the mean sleep time is less than 7 hours (rejecting the false null hypothesis) when the population’s true mean sleep time is equal to 6.0 hours.

Value of  β

$\beta$The value of $\beta$ is computed using the following formula:

$$\begin{gathered} \text{Power}=1-\beta \\ \beta =1-\text{Power} \end{gathered}$$

Therefore, the value of is equal to:

$$\begin{gathered} \beta =1-\text{Power} \\ =1-0.4943 \\ =0.5057 \end{gathered}$$

Thus, the value of $\beta$ is equal to 0.5057.

Interpretation of the value of  β

The value of $\beta$ is the value of the Type II error which is the probability of failing to reject the null hypothesis when it is actually false.

In simple words, $\beta$ refers to the probability of making an incorrect conclusion about the stated claim.

For the given claim, the value ofrole="math" localid="1649069586410" $\beta$represents the probability of rejecting the claim that the mean sleep time is less than 7 hours when the true population’s mean sleep time is actually equal to 6.0 hours.